Problem: Nadia works out for $\frac{2}{3}$ of an hour every day. To keep her exercise routines interesting, she includes different types of exercises, such as push-ups and jumping jacks, in each workout. If each type of exercise takes $\frac{2}{9}$ of an hour, how many different types of exercise can Nadia do in each workout?
To find out how many types of exercise Nadia could do in each workout, divide the total amount of exercise time ( $\frac{2}{3}$ of an hour) by the amount of time each exercise type takes ( $\frac{2}{9}$ of an hour). $ \dfrac{{\dfrac{2}{3} \text{ hour}}} {{\dfrac{2}{9} \text{ hour per exercise}}} = {\text{ number of exercises}} $ Dividing by a fraction is the same as multiplying by the reciprocal. The reciprocal of ${\dfrac{2}{9} \text{ hour per exercise}}$ is ${\dfrac{9}{2} \text{ exercises per hour}}$ $ {\dfrac{2}{3}\text{ hour}} \times {\dfrac{9}{2} \text{ exercises per hour}} = {\text{ number of exercises}} $ $ \dfrac{{2} \cdot {9}} {{3} \cdot {2}} = {\text{ number of exercises}} $ Reduce terms with common factors by dividing the $2$ in the numerator and the $2$ in the denominator by $2$ $ \dfrac{{\cancel{2}^{1}} \cdot {9}} {{3} \cdot {\cancel{2}^{1}}} = {\text{ number of exercises}} $ Reduce terms with common factors by dividing the $9$ in the numerator and the $3$ in the denominator by $3$ $ \dfrac{{1} \cdot {\cancel{9}^{3}}} {{\cancel{3}^{1}} \cdot {1}} = {\text{ number of exercises}} $ Simplify: $ \dfrac{{1} \cdot {3}} {{1} \cdot {1}} = {3} $ Nadia can do 3 different types of exercise per workout.